| name | hamiltonian-simulation |
| description | Quantum Hamiltonian simulation methods for approximating time evolution e^{-iHt}. Includes Trotter-Suzuki decomposition, QDrift randomized sampling, Cartan decomposition, QSP-HS polynomial spectral transformation, and Taylor series LCU expansion. |
Hamiltonian Simulation
Approximate the time-evolution operator $e^{-iHt}$ for a given Hermitian matrix $H$.
Available Methods
1. Trotter-Suzuki Decomposition
See reference: ./trotter/SKILL.md
Deterministic product formula. Decomposes $e^{-iHt}$ into a sequence of simpler exponentials.
- Key parameters:
order (Suzuki order), steps (number of Trotter steps).
2. QDrift (Randomized)
See reference: ./qdrift/SKILL.md
Stochastic channel simulation via $\lambda$-weighted random sampling of Pauli terms.
- Key parameter:
steps (number of random samples).
3. Cartan Decomposition
See reference: ./cartan/SKILL.md
Structural decomposition via Lie algebra splitting $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Iterates a Lax flow to build a circuit of the form $K \cdot e^{-i\eta} \cdot K^\dagger$.
- Key parameters:
lr (learning rate), max_steps (hard cap on Lax update steps), reps (number of repetitions).
4. QSP-HS (Quantum Signal Processing for Hamiltonian Simulation)
See reference: ./qsp/SKILL.md
Use this route only when the user asks for Hamiltonian simulation or time evolution with QSP. This method block-encodes the Hamiltonian and applies polynomial spectral transformations via interleaved signal-processing rotations. It approximates $\cos(tH)$ and $\sin(tH)$ as Chebyshev series and merges them via LCU.
- Key parameters:
degree (upper bound on polynomial degree per time slice), beta (block-encoding scaling factor).
For a basic single-qubit QSP demo, scalar signal variable x, phase-factor optimization, or approximating cos(t*x) at a point, route to ../linear-systems/quantum-signal-processing/SKILL.md instead.
5. Taylor Series (LCU)
See reference: ./taylor/SKILL.md
Truncates the Taylor series of $e^{-iHt}$ and implements the result as a Linear Combination of Unitaries over Pauli string products. Applies adaptive time-slicing to keep the per-slice spectral weight small.
- Key parameters:
degree (Taylor truncation order, capped at 15).
Method Selection Guidance
This top-level skill is an index and routing guide. For executable examples, open the corresponding method-specific SKILL.md.
Conceptual dispatch logic:
if method == "trotter":
use ./trotter/SKILL.md
elif method == "qdrift":
use ./qdrift/SKILL.md
elif method == "cartan-lax":
use ./cartan/SKILL.md
elif method == "qsp" or method == "qsp-hs":
use ./qsp/SKILL.md
elif method == "taylor":
use ./taylor/SKILL.md
| Method | method string | Extra kwargs |
|---|
| Trotter | 'trotter' | order, steps |
| QDrift | 'qdrift' | steps |
| Cartan | 'cartan-lax' | lr, max_steps, reps |
| QSP-HS | 'qsp' / 'qsp-hs' | degree, beta |
| Taylor | 'taylor' | degree |
The returned object exposes lazy properties: circuit, evolution_result, total_error.
Prerequisites
- Basic linear algebra (Hermitian matrices, matrix exponential).
- High-level understanding of Hamiltonian simulation goals.
- NumPy for matrix construction.
